Mixed Second-order Derivatives Research Articles

EN General thermodynamic relations for thermal variables formed by first-order partial derivatives and not mixed second-order derivatives

The theory of differential equations is a powerful tool necessary for constructing mathematical models of various applied problems and their solution. The study and analysis of the general differential relations between the derivatives of the thermodynamic functions of substances is a new direction in the theory of their thermodynamic properties. This research is the development of thermodynamics associated with the establishment and use of differential equations. This paper presents two new thermodynamic relationships between thermal variables pressure, absolute temperature, and specific volume. They relate first-order partial derivatives with unmixed second-order derivatives for these variables. Revealing new general relations of thermodynamics made it possible to formulate and solve a number of problems, among which we can note: creation of algorithms for calculating second-order mixed derivatives for a given thermal equation of state; analysis of the features of the second mixed derivatives at the critical point of pure substance; investigations of the general thermodynamic equations with entropy. The procedure for establishing new equations is based on the use of a rigorous mathematical apparatus and Maxwell's differential equations. Therefore, they are of a general nature, that is, they are valid for any pure substance or any mixture of fixed composition. They are also applicable to any model of the equation of state of these substances. With a formal approach to the concept of the term "equation of state", the new relations themselves can be considered as the equation of state of substance. The obtained new relations can be used to develop the theory of the thermodynamic properties of substances. In particular, in this work it is shown that the ratios of two partial derivatives of pressure over the volume, which are zero critical conditions, have no singularities at the critical point itself

Two stochastic algorithms for solving elastostatics problems governed by the Lamé equation

Abstract In this paper, we construct stochastic simulation algorithms for solving an elastostatics problem governed by the Lamé equation. Two different stochastic simulation methods are suggested: (1) a method based on a random walk on spheres, which is iteratively applied to anisotropic diffusion equations that are related through the mixed second-order derivatives (this method is meshless and can be applied to boundary value problems for complicated domains); (2) a randomized algorithm for solving large systems of linear algebraic equations that is the core of this method. It needs a mesh formation, but even for very fine grids, the algorithm shows a high efficiency. Both methods are scalable and can be easily parallelized.

Идентификация квадратичного ядра уравнения Вольтерра для моделирования нелинейных динамических систем

In the last two decades, integral models have been used to describe the dynamics of stationary nonlinear systems in which the terms of the Volterra series are the kernels. The most commonly used are the linear term (the impulse transition function depends on one variable) and the quadratic term (depending on two variables). An active experiment in which a special combination of rectangular pulses is fed to the input of the system is carried out to select two of its components in the output signal of the identified system - the output of the linear "subsystem" and the output of the "quadratic" subsystem. After isolating the output of the "quadratic" subsystem, the identification of the quadratic term of the Volterra series is reduced to solving a two-dimensional integral equation of the first kind. In the literature, inversion formulas are given in which the quadratic kernel function is obtained as a result of arithmetic operations with second-order derivatives of the output signal. Differentiation of functions is an incorrectly posed problem, when small errors in the assignment of a function (measurement noise) cause large errors in derivatives (especially in second-order derivatives). The paper proposes the use of smoothing cubic splines for stable calculation of derivatives. To calculate the mixed second-order derivative, a spline with two variables is built - a smoothing bicubic spline. The main problem that arises in practice when processing the data of a real experiment is the selection of a smoothing parameter, on the value of which a smoothing error of noisy data depends. As a rule, the value of the variance of the measurement noise is unknown in the experiment. Therefore, in this work, it is proposed to use an algorithm based on the L-curve method to select the smoothing parameter in the constructed splines (especially in the bicubic one), which does not require setting the variance of the measurement noise. The proposed identification algorithm has a high computational efficiency. The performed computational experiment showed a small methodical error (about 1%) and good resistance to noise in measurements of the output signals of the identified system. To reduce the random component of the identification error, it is proposed to use post-processing with a local-spatial compound filter.

Fluctuation Theory in Chemical Kinetics

In this research, we study the stability properties of chemical reactions of arbitrary orders. In a given chemical experiment, we focus on the formation of a chemical equilibrium by optimizing the reaction rate. Under infinitesimal simultaneous variations of the concentrations of reacting species, the binary component equilibrium is achieved when either one of the orders or concentrations of reactants vanishes. The chemical concentration capacities of the components are calculated to describe the local stability of the equilibrium. The correlation between the components is obtained as the mixed second-order derivative of the rate with respect to concentrations. The global stability analysis is performed by introducing a symmetric matrix with its diagonal components as the chemical capacities and off-diagonal components as the local correlation. We find that the local chemical stability requires the orders of the reactants to be either negative or larger than unity. The corresponding global stability requires the positivity of a cubic factor over the orders of the reactants. In short, our consideration illustrates how a chemical reaction takes place by attaining its activation state and asymptotically approaches the equilibrium when two components are mixed with arbitrary orders. Qualitative discussions are provided to support our analysis towards the formation of an optimized equilibrium. Finally, along with future directions, we discuss verification of our model towards the formation of carbon-based reactions, formation of organic/inorganic chemical equilibria and catalytic oxidation of C O − H 2 mixtures in presence of Pt.

Essentially High-Order Compact Schemes with Application to Stochastic Volatility Models on Non-Uniform Grids

We present high-order compact schemes for a linear second-order parabolic partial differential equation (PDE) with mixed second-order derivative terms in two spatial dimensions. The schemes are applied to option pricing PDE for a family of stochastic volatility models. We use a non-uniform grid with more grid-points around the strike price. The schemes are fourth-order accurate in space and second-order accurate in time for vanishing correlation. In our numerical convergence study we achieve fourth-order accuracy also for non-zero correlation. A combination of Crank-Nicolson and BDF-4 discretisation is applied in time. Numerical examples confirm that a standard, second-order finite difference scheme is significantly outperformed.

High-Order Compact Schemes for Parabolic Problems with Mixed Derivatives in Multiple Space Dimensions

We present a high-order compact finite difference approach for a class of parabolic partial differential equations with time- and space-dependent coefficients as well as with mixed second-order derivative terms in $n$ spatial dimensions. Problems of this type arise frequently in computational fluid dynamics and computational finance. We derive general conditions on the coefficients which allow us to obtain a high-order compact scheme which is fourth-order accurate in space and second-order accurate in time. Moreover, we perform a thorough von Neumann stability analysis of the Cauchy problem in two and three spatial dimensions for vanishing mixed derivative terms, and also give partial results for the general case. The results suggest unconditional stability of the scheme. As an application example we consider the pricing of European power put basket options in the multidimensional Black--Scholes model for two and three underlying assets. Due to the low regularity of typical initial conditions we employ the smoothing operators of Kreiss, Thomee, and Widlund [Comm. Pure Appl. Math., 23 (1970), pp. 241--259] to ensure high-order convergence of the approximations of the smoothed problem to the true solution.

On a boundary-value problem with data on noncharacteristic intersecting lines for systems of hyperbolic equations with mixed derivative

We consider a boundary-value problem with data located on noncharacteristic intersecting lines for systems of hyperbolic equations with mixed second-order derivative. Using the method of additional parameters, we establish the existence of a unique classical solution of this problem and develop a procedure for finding this solution.

Simple and fast calculation of the second-order gradients for globalized dual heuristic dynamic programming in neural networks.

We derive an algorithm to exactly calculate the mixed second-order derivatives of a neural network's output with respect to its input vector and weight vector. This is necessary for the adaptive dynamic programming (ADP) algorithms globalized dual heuristic programming (GDHP) and value-gradient learning. The algorithm calculates the inner product of this second-order matrix with a given fixed vector in a time that is linear in the number of weights in the neural network. We use a "forward accumulation" of the derivative calculations which produces a much more elegant and easy-to-implement solution than has previously been published for this task. In doing so, the algorithm makes GDHP simple to implement and efficient, bridging the gap between the widely used DHP and GDHP ADP methods.

Stability analysis for acoustic wave propagation in tilted TI media by finite differences

SUMMARY Several papers in recent years have reported instabilities in P-wave modelling, based on an acoustic approximation, for inhomogeneous transversely isotropic media with tilted symmetry axis (TTI media). In particular, instabilities tend to occur if the axis of symmetry varies rapidly in combination with strong contrasts of medium parameters, which is typically the case at the foot of a steeply dipping salt flank. In a recent paper, we have proposed and demonstrated a P-wave modelling approach for TTI media, based on rotated stress and strain tensors, in which the wave equations reduce to a coupled set of two second-order partial differential equations for two scalar stress components: a normal component along the variable axis of symmetry and a lateral component of stress in the plane perpendicular to that axis. Spatially constant density is assumed in this approach. A numerical discretization scheme was proposed which uses discrete second-derivative operators for the non-mixed second-order derivatives in the wave equations, and combined first-derivative operators for the mixed second-order derivatives. This paper provides a complete and rigorous stability analysis, assuming a uniformly sampled grid. Although the spatial discretization operator for the TTI acoustic wave equation is not self-adjoint, this operator still defines a complete basis of eigenfunctions of the solution space, provided that the solution space is somewhat restricted at locations where the medium is elliptically anisotropic. First, a stability analysis is given for a discretization scheme, which is purely based on first-derivative operators. It is shown that the coefficients of the central difference operators should satisfy certain conditions. In view of numerical artefacts, such a discretization scheme is not attractive, and the non-mixed second-order derivatives of the wave equation are discretized directly by second-derivative operators. It is shown that this modification preserves stability, provided that the central difference operators of the second-order derivatives dominate over the twice applied operators of the first-order derivatives. In practice, it turns out that this is almost the case. Stability of the desired discretization scheme is enforced by slightly weighting down the mixed second-order derivatives in the wave equation. This has a minor, practically negligible, effect on the kinematics of wave propagation. Finally, it is shown that non-reflecting boundary conditions, enforced by applying a taper at the boundaries of the grid, do not harm the stability of the discretization scheme.

Rototranslational sum rules for electromagnetic hypershielding at the nuclei and related atomic Cartesian derivatives of the optical rotatory power

Two molecular properties, the nuclear electromagnetic hypershielding (psi(gamma,alphabeta) ('I)) and the gradient of the electric dipole-magnetic dipole polarizability (nabla(Igamma)G(alphabeta) (')), have been calculated using the time-dependent Hartree-Fock method. Provided the Hellmann-Feynman theorem is satisfied, these quantities are equivalent and are related through the nabla(Igamma)G(alphabeta) (')=eZ(I)psi(gamma,alphabeta) ('I) relation, where Z(I) is the atomic number of atom I and e the magnitude of the electron charge. In such a case, the determination of the nuclear electromagnetic hypershielding presents the computational advantage over the evaluation of the gradient of G(alphabeta) (') of requiring only the knowledge of nine mixed second-order derivatives of the density matrix with respect to both electric and magnetic fields (D(alpha,beta)(-omega,omega)) instead of the 3N (N is the number of atoms) derivatives of the density matrix with respect to the Cartesian coordinates (D(Igamma)). It is shown here for the H(2)O(2) molecule that very large basis sets such as the aug-cc-pVQZ or the R12 basis are required to satisfy the Hellmann-Feynman theorem. These basis set requirements have been substantiated by considering the corresponding rototranslational sum rules. The origin dependence of the rototranslational sum rules for the gradient of G(alphabeta) (') has then been theoretically described and verified for the H(2)O(2) molecule.

Separation of components from impulses in reassigned spectrograms

Two computational methods for pruning a reassigned spectrogram to show only quasisinusoidal components, or only impulses, or both, are presented mathematically and provided with step-by-step algorithms. Both methods compute the second-order mixed partial derivative of the short-time Fourier transform phase, and rely on the conditions that components and impulses are each well-represented by reassigned spectrographic points possessing particular values of this derivative. This use of the mixed second-order derivative was introduced by Nelson [J. Acoust. Soc. Am. 110, 2575-2592 (2001)] but here our goals are to completely describe the computation of this derivative in a way that highlights the relations to the two most influential methods of computing a reassigned spectrogram, and also to demonstrate the utility of this technique for plotting spectrograms showing line components or impulses while excluding most other points. When applied to speech signals, vocal tract resonances (formants) or glottal pulsations can be effectively isolated in expanded views of the phonation process.

Envelope equation for nonlinear transverse waves in a warm two-fluid plasma

An equation describing the nonlinear development of an envelope of weakly dispersive transverse waves propagating along the ambient magnetic field in a warm two-fluid plasma is derived by using the Fourier transformation method. This equation contains three nonlinear terms, a cubic, a derivative of cubic, and a quadratic multiplied by a derivative term. For the case of small wave amplitude, the equation is of non-Schrödinger type that includes a linear term involved with mixed second-order derivatives (space and time), in addition to the usual two terms of the Schrödinger-type equation. This mixed derivative term arises from the inclusion of the displacement current in the formulation. The nonlinear terms in the derived equation explicitly contain a smallness parameter which vanishes in the limit of zero dispersion. The low frequency limit of the equation is compared with the standard derivative nonlinear Schrödinger equation (DNLS), and differences are discussed. The new equation predicts that the condition for the modulational instability for the low frequency circularly polarized waves does not depend on the sense of polarization, a result that differs from the usual DNLS equation. The high and intermediate frequency limits of the equation are also considered.

Dynamical matrices, Born effective charges, dielectric permittivity tensors, and interatomic force constants from density-functional perturbation theory

Starting from the knowledge of first-order changes of wave functions and density with respect to small atomic displacements or infinitesimal homogeneous electric fields within the density-functional theory, we write the expressions for the diagonal or mixed second-order derivatives of the total energy with respect to these perturbations: dynamical matrices for different wave vectors, Born effective-charge tensors and electronic dielectric permittivity tensors. Interatomic force constants and the phonon-band structure are then obtained by computing the Fourier transform of dynamical matrices on a regular mesh of wave vectors, with an eventual, separate treatment of the long-range dipole-dipole interaction. The same ingredients also allow one to compute the low-frequency response of the crystal to homogeneous electric fields.

Theory of propagation of elastic waves in nonhomogeneous media

1. The effective tensor operator of the wave equation of ideally disordered composites is an integrodifferential tensor operator containing terras with second derivatives with respect to the coordinates and time and mixed second-order derivatives. The kernels of the spectral operators on which the named differential operators act are components of tensors of the fourth, second, and third rank. Their dependence on the coordinates is determined by the specific form of the binary correlation functions of the material constants. 2. General formulas for the scattering coefficient and phase and group velocities of propagation of longitudinal and transverse waves can be obtained by using the Burre approximation, an exponential coordinate relation for the binary correlation functions, and the notion that the mean field in the composite, in the form of a monochromatic wave, is characterized by the effective wave vector. 3. Analytic expressions were obtained for one longwave and two shortwave asymptotes of the investigated parameters. It was shown that these asymptotes coincide with the literature data in the longwave case and are confirmed by experimental results in the shortwave case.